Anna Freud, daughter of Sigmund Freud and an eminent psychoanalyst in her own right, died at her London home today at the age of 86, her family said. ''I started out as an elementary-school teacher. I changed from that to the field of analysis and child analysis. From then on, I moved constantly back and forth, from the theoretical study of these problems to their practical application. I agree that one has to have special luck to do this, and that most people do not have this.'' This was Anna Freud's self-effacing summary of a long and productive life. It was true, of course, so far as it went; but it obscured the fact that Miss Freud, eminent in her own right, virtually invented the systematic study of the emotional and mental life of the child and elaborated it in 50 years of observation, discussion and writing. She also understated her powerful role as the guardian and elucidator of her father's revolutionary doctrine on the nature of the human psyche and its behavioral implications. Although Miss Freud was not alone in altering established perceptions of child development, she was among the first to discern the significance of the secret wishes and fears of the young, their phobias, obsessions and nightmares. Earlier it was generally assumed that children were arbitrarily motivated and that discipline was the surest path to healthy development. Miss Freud suggested that a child developed emotionally through a number of stages - the oral, anal, urethral and phallic sexual stages that Sigmund Freud had outlined from circumstantial evidence. His argument for the existence of these stages was often faulted because it had been based on the recollections of adults, and troubled ones at that. Dr. Robert Coles of Harvard was among many who credited Miss Freud with proving one of her father's basic hypotheses; but there were others who doubted that her proofs were more than self-serving, or who challenged Freudianism altogether. Even doubters, however, seemed willing to agree that Miss Freud had indeed contributed to an understanding of the ways children find to get along with their parents, siblings and friends. Two of her books are counted essential for those who work with children -''The Psychoanalytic Treatment of Children'' and ''Psychoanalysis for Teachers and Parents.'' Both were published in the late 1920's and were written with what Dr. Coles called ''a modesty of spirit.'' Another of her significant books was ''War and Children,'' written in collaboration with a friend, Dorothy T. Burlingham, and based on case studies of the effects of World War II bombing on British children. It upset preconceived notions of children's reactions. ''General sympathy has been aroused by the idea that little children, all innocence, should come into close contact with the horrors of war,'' the authors wrote, and then showed by example that children were not at all saddened by the sight of destruction. ''Love for the parents is so great that it is a far greater shock for a child to be suddenly separated from its mother than to have a house collapse on top of it,'' the book reported. ''It is surprising how little interest children show in sirens, bombs, guns and allclears.'' Miss Freud and Miss Burlingham concluded that the world of the small child is pivoted on the mother and that the child has little if any instinctive horror of combat, blood or destruction. Miss Freud, the youngest of six children, was born in Vienna on Dec. 3, 1895, and was reared in comfortable bourgeois circumstances. Close to her father, who was in his late 40's when she was growing up, Anna was keenly aware of the patriarchal figure that her blackbearded father cut, of his rather rigid routines and of his happy sense of fun. She was aware, too, of her father's ostracism by the anti-Semitic scientific establishment in Vienna despite his standing as a neurologist, a hostility so harsh that his masterwork, ''The Interpretation of Dreams,'' was ignored when it was published in 1900. It took eight years to sell 600 copies of the book. Reticent about her adolescence, as she also was about most aspects of her life, Miss Freud was unwilling to tell how or why she was drawn to psychoanalysis as a life absorption; friends said that admiration for her father was at the bottom of it, and she undertook to comprehend his theories. He, in turn, psychoanalyzed her - an unorthodox procedure. In her late 20's, Miss Freud became a psychoanalyst without acquiring a medical degree, a most unusual circumstance. Her first patients were adults, but she quickly shifted to children and was the leader of a small group of analysts who met regularly to discuss the problems of children. In the group were Dr. Marianne Kris and Dr. Erik H. Erikson. Co-workers remember Miss Freud as having even then some of her father's commanding presence - ''a certain tilt of the head and a fire in the eyes.'' Robert Sussman Stewart, a New York editor who wrote about psychoanalysis, said of Miss Freud: ''She also had her father's ability, though on a smaller scale, to rally others or, more to the point, to generate disciples when she needed them. But if she was formidable in these respects, there were ways in which she could be surprisingly modest and self-effacing: with difficult children she showed inexhaustible patience; and to the work of the few she considered her colleagues she gave all manner of attention, often at the expense of her own productivity.'' Her papers and articles from these and other years were later collected in a multivolume ''The Writings of Anna Freud,'' and issued in the United States by International Universities Press. One of her books, ''The Ego and Mechanism of Defense,'' published in English in 1937, was described by Dr. Coles as ''one of the most important psychoanalytic books ever written.'' Miss Freud was with her father in Vienna when the Germans took over Austria in 1938 and held the Freuds for ransom. The family was permitted to go to Britain after the payment of money and after Freud had signed a statement that he had been well treated. To the statement he appended the ironic note, ''I can recommend the Gestapo to anyone.'' In London, Miss Freud helped to establish her father and mother at 20 Marsefield Gardens, in the Hampstead area, where Freud died in 1939. The house was the site of the Hampstead Child Therapy Clinic, which she established for the study of children and the training of analysts. She was its director until her death. She ruled over the clinic with a tactful but firm hand, selecting its staff and approving both the children admitted for study and the analysts who would be trained. She herself lived in the house where her father died, and in plain circumstances, considering her fame. A sunny, pleasant, memento-filled room on the second floor was her study. She often greeted visitors there, and would talk patiently with them if she felt they were not too inquisitive about her. Understandably, she was suspicious (or fearful) lest her words be distorted, for she felt that vulgarization had contributed to misunderstandings about the humanist core of psychoanalysis. Her anger with Vienna was almost implacable. Not only had the Viennese scorned her father and refused to protest the German outrages against him, but they had never, in the years since World War II, honored his memory in any way. She relented only in 1971 for a meeting of the International Psychoanalytic Association and only after Vienna had officially expressed regrets for its treatment of the Freuds. Earlier, she had said half-jokingly that she would visit Vienna only if the city provided her with a Lippizaner horse on which to ride through the gates. As it was, the city gave her a special reception, at which she waltzed and sipped wine. Appearing at the association's sessions, she was moved almost to tears by an ovation from the delegates. Then a seemingly frail woman of 75 with frizzy gray hair and dancing eyes, she had a constitution of iron and a mind that missed nothing. She summed up the long and complex sessions in an hour's extemporaneous speech that was remarkable for its organization, pithiness and coherence. She was undismayed by the minuscule number of psychoanalysts in the world - about 3,000 - and the disregard that many psychiatrists had for Freudianism on scientific grounds. Remarking on the differences among Freudians over aggression, she asked, ''But who should arrive at valid conclusions if not we?'' With a firm grasp of her father's work, she was the accepted explicator of his texts and was listened to accordingly. However, she did not wield absolute authority over child psychoanalysis. Her rival (and bete noire) was the late Melanie Klein, the daughter of a Viennese physician, who was encouraged by Freud himself to study children. Whereas Mrs. Klein believed that a child was born to be paranoiac and depressive, Miss Freud saw the earliest years of infancy as a prologue to greater maturity. Her therapy stressed protective, supportive and educational attitudes toward the child patient. Among other things, she suggested how the classic modalities of adult psychoanalysis - transference, resistance, motivation and free association - could be utilized with children 4 years old and upward. Miss Freud, of course, did not advocate analysis for every disturbed child. Many children pass through developmental upsets, and their problems are transitory, she believed, so that candidates for her therapy were largely the pathologically sick. Although Miss Freud was loath to talk about herself, she was eager to take center stage in lecturing about children, especially with regard to protecting their natural rights. To this end, she held seminars at the Yale Law School that dealt with penal codes and children, and at the Yale Child Study Center on related subjects. Miss Freud felt certain that her work (and by that she meant the work of her Hampstead center) fulfilled her life and benefited children generally. She was not so sanguine, however, about the immediate future of psychoanalysis in Western culture. ''Young people now are not interested in man's struggle against himself, but in man's struggle against society,'' she said. ''They see that what psychoanalysis may lead to is adaptation to society, and that is the last thing they have in mind.'' ''But eventually,'' she went on, ''psychoanalysis will grow and take us deeper and deeper into life.''